Conditional quantum one-time pad

Understand

Suppose that Alice and Bob are located in distant laboratories, which are connected by an ideal quantum channel.

• Suppose further that they share many copies of a quantum state $\rho_{ABE}$, such that Alice possesses the $A$ systems and Bob the $BE$ systems.
• In our model, there is an identifiable part of Bob's laboratory that is insecure: a third party named Eve has infiltrated Bob's laboratory and gained control of the $E$ systems.
• Alice, knowing this, would like use their shared state and the ideal quantum channel to communicate a message in such a way that Bob, who has access to the whole of his laboratory ($BE$ systems), can decode it, while Eve, who has access only to a sector of Bob's laboratory ($E$ systems) and the ideal quantum channel connecting Alice to Bob, cannot learn anything about Alice's transmitted message.